Monoids Connected with Euler's Diophantine Equation
نویسندگان
چکیده
منابع مشابه
A Binomial Diophantine Equation
following result. THEOREM 1. The only (n,m)eZ with n^2 and m5=4 satisfying © = ( 7 ) a r e {n> m)=(2> 4)> (6> 6)> and (21> Our binomial diophantine equation represents an elliptic curve, since it can be rewritten as a quartic polynomial being a square. Indeed, on putting u = 2/i 1 and v = 2m 3, we see at once that Theorem 1 follows from the following result. THEOREM 2. The only (u, v) e Z with ...
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= c for some integers a, b, c with ab 6= 0, has only finitely many integer solutions. Stoll & Tichy proved more generally that if a, b, c ∈ Q and ab 6= 0, then for m > n ≥ 3, the above equation has only finitely many integral solutions x, y. Independently, Rakaczki established a more precise finiteness result on this binomial equation and extended this result to more general equations (see Acta...
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If a, b and n are positive integers with b ≥ a and n ≥ 3, then the equation of the title possesses at most one solution in positive integers x and y, with the possible exceptions of (a, b, n) satisfying b = a + 1, 2 ≤ a ≤ min{0.3n, 83} and 17 ≤ n ≤ 347. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometri...
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ژورنال
عنوان ژورنال: Missouri Journal of Mathematical Sciences
سال: 1998
ISSN: 0899-6180
DOI: 10.35834/1998/1002091